The Intertemporal Keynesian Cross

The Intertemporal Keynesian Cross Adrien Auclert Matthew Rognlie Ludwig Straub February 14, 2018 PRELIMINARY AND INCOMPLETE,PLEASE DO NOT CIRCULATE Abstract This paper develops a novel approach to analyzing the transmission of shocks and policies in many existing macroeconomic models with nominal rigidities. Our approach is centered around a network representation of agents’ spending patterns: nodes are goods markets at different times, and flows between nodes are agents’ marginal propensities to spend income earned in one node on another one. Since, in general equilibrium, one agent’s spending is another agent’s income, equilibrium demand in each node is described by a recursive equation with a special structure, which we call the intertemporal Keynesian cross (IKC). Each solution to the IKC corresponds to an equilibrium of the model, and the direction of indeterminacy is given by the network’s eigenvector centrality measure. We use results from Markov chain potential theory to tightly characterize all solutions. In particular, we derive (a) a generalized Taylor principle to ensure bounded equilibrium determinacy; (b) how most shocks do not affect the net present value of aggregate spending in partial equilibrium and nevertheless do so in general equilibrium; (c) when heterogeneity matters for the aggregate effect of monetary and fiscal policy. We demonstrate the power of our approach in the context of a quantitative Bewley-Huggett-Aiyagari economy for fiscal and monetary policy. 1 Introduction One of the most important questions in macroeconomics is that of how shocks are propagated and amplified in general equilibrium. Recently, shocks that have received particular attention are those that affect households in heterogeneous ways—such as tax rebates, changes in monetary policy, falls in house prices, credit crunches, or increases in economic uncertainty or income in- equality. New empirical evidence has been brought to bear, and new decision-theoretic models have been developed, to quantify and understand the causal effect of these shocks on aggregate consumption.1 Yet both cross-sectional empirical evidence and decision-theory models deliver 1For empirical evidence using cross-sectional variation, see Johnson, Parker and Souleles(2006) for tax rebates, on or Fagereng, Holm and Natvik(2016) on unanticipated income shocks. For mostly partial equilibrium decision-theory 1 partial equilibrium estimates of aggregate effects. A significant question that remains is how partial equilibrium effects translate into general equilibrium outcomes, once markets clear and the endogenous responses of monetary and fiscal policy are taken into account. This is the question we study in this paper. It is intuitive that there should be commonalities in the partial-to-general equilibrium adjustment mechanism. In a closed economy, whether consumption falls because of a fall in house prices, a credit crunch, or a fiscal retrenchment, common mechanisms are likely to operate to restore equality between equilibrium demand and output. One important such adjustment mechanism, highlighted by Keynes(1936), is the feedback between consumption and incomes: as consumption falls, incomes across the population falls, triggering further adjustments in consumption. Modern models used for monetary and fiscal policy analysis tend to emphasize other mechanisms instead: those that arise from the endogenous reaction of policy to the fall in economic activity. These models typically feature representative agents who smooth their consumption responses to income shocks over their infinite lifetime, rendering the consumption-income feedback quantitatively irrelevant. But a newer vintage of models incorporates much larger marginal propensities to consume out of transitory income shocks, which empirical evidence has consis- tently shown to be an important feature of the data. In these models, consumption-income feedbacks have the potential to play a much more important role for general equilibrium adjustment. We show that, in a broad class of models with nominal rigidities, the question of how to go from fundamental shocks to equilibrium outcomes is answered by the solution to one simple recursive equation, dY=¶Y+M dY (1) · In (1), the vector dY represents the general equilibrium impulse response to the shock–the change in the time path of output induced by the shock, which is the ultimate object of interest to macroeconomic analysis. ¶Y represents the partial equilibrium impulse response, whose definition we establish precisely below, and which in our benchmark case corresponds to the effect of the shock holding interest rates and quantities constant, as in the aforementioned literature. Finally, M is a matrix of general equilibrium feedbacks, encapsulating the response of private individuals to changes in incomes and interest rates, as well as endogenous policy responses, notably that of monetary policy. In a benchmark case in which real interest rates are held constant by monetary policy (’constant-r’), the entries of the M matrix correspond to households’ average marginal propensities to consume, at each date, for income received at other dates, weighted by the inci- dence of income across the population. This renders the M matrix potentially observable directly, just as ¶Y can be, opening the promise of conducting general equilibrium analysis using a suffi- cient statistic approach under minimal model assumptions. Keynes(1936) proposed an answer similar to our equation (1), focusing exclusively on the income-consumption feedback mechanism and the immediate response of output to shocks. In models, see Auclert(2017) on the effects of monetary policy, Kaplan and Violante(2014) on tax rebates, and Berger, Guerrieri, Lorenzoni and Vavra(2015) on house prices. 2 his formulation, dY = ¶Y + MPC dY (2) · where ¶Y is the impulse to aggregate demand, dY is the equilibrium change in output, and MPC is the aggregate marginal propensity to consume out of income. Equation (2) can be solved to obtain dY = (1 MPC) 1¶Y, where (1 MPC) 1 = 1 + MPC + MPC2 + . is called the multiplier − − − − and reflects the accumulated consumption feedback amplifying the original impulse. Although this traditional ‘Keynesian cross’ embodies a useful intuition, it has a number of weaknesses. It is static, not dynamic. It does not require that budget constraints be satisfied: an impulse to demand from government spending comes out of thin air, rather than being offset by taxation or a cut in spending at some other date. It does not directly correspond to any standard, microfounded model. It does not give a role to monetary policy. It assumes that the response of the economy is determinate—an assumption modern monetary theory has shown is only guaranteed under particular types of policy rules. We show that, nevertheless, going back to these intuitions is useful to understand modern business cycle theory, and furthermore the parallel delivers novel results that had escaped the literature until now. First, when monetary policy maintains the real interest rate constant, the M matrix in (1) corresponds to a weighted average of marginal propensities to consume. This captures the static consumption-income feedbacks emphasized by Keynes, as well as richer intertemporal interactions: for instance, some income earned in period 1 will be spent in period 2, from which some income will be spent again in period 1. Budget constraints also impose a number of important modifications to the static view. First, it is not generally possible to simply solve for dY in (1) without further considerations. The crucial observation here is that all income is eventually consumed in net present value terms–the dynamic M matrix has an eigenvalue of 1, making inversion of I M impossible in general. On the other hand, partial equilibrium shocks − on their own do not create present value. In our dynamic case, we show that equation (1) can be solved, but that it generally has multiple solutions.2 This indeterminacy is inherent to models with nominal rigidities, and for the first time, we are sharply able to characterize it, by showing how it relates to properties of the M matrix. To understand the amplification mechanisms inherent to our model is useful to think of time periods as nodes of a network, as in figure1. Each new unit of income generated at a given node is spent, partly on itself and partly on every other node, according to relationships given by the matrix M. This round of spending generates additional income at each node, which is again spent according to the same pattern, and so on. The final outcome for the distribution of income across nodes is our main object of interest. It is the general equilibrium effect on consumption after the intertemporal Keynesian cross has run its course. If the matrix M is written in net present value units—as we generally do—then the fact that all income is spent in net present value terms means that M is a left-stochastic matrix, with columns summing to one. It can then be interpreted as 2Intutitively, in the scalar version (2), budget constraints impose that ¶Y = 0 and MPC = 1, so any value of dY is a solution. 3 M ,0 · M0,0 M , · · M1,0 M1, · t = 0 t = 1 ··· M0,1 M ,1 · M1,1 M0, · Figure 1: Interpretating the economy as a network the transition matrix for a Markov chain, with dimension equal to the number of periods of the model. The stationary distribution of this Markov chain, if it exists, delivers the dimension of indeterminacy: if agents expect aggregate income to increase in proportion to this distribution over time, then their collective consumption responses are such that it sustain exactly this additional increase in income. Sometimes, however, when = ¥, the only stationary ‘distributions’ diverge T with time—an outcome that is natural to rule out, as the literature typically does. In this case, equilibrium is uniquely pinned down. We first show that, under constant-r monetary policy, whether or not the equilibrium is determinate depends on the economy’s M matrix.

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